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# hw28solutions - Math 521 Advanced Calculus I Instructor J...

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Math 521 - Advanced Calculus I Instructor: J. Metcalfe Due: April 23, 2010 Assignment 28 1. Use the following argument to prove the Substitution Theorem (Theorem 7.3.8). Deﬁne F ( u ) = R u φ ( α ) f ( x ) dx for u I , and H ( t ) = F ( φ ( t )) for t J . Show that H 0 ( t ) = f ( φ ( t )) φ 0 ( t ) for t J and that Z φ ( β ) φ ( α ) f ( x ) dx = F ( φ ( β )) = H ( β ) = Z β α f ( φ ( t )) φ 0 ( t ) dt. By the chain rule, we have H 0 ( t ) = F 0 ( φ ( t )) φ 0 ( t ) and by the Fundamental Theorem of Calculus, H 0 ( t ) = f ( φ ( t )) φ 0 ( t ) . By deﬁnition H ( β ) = F ( φ ( β )) = Z φ ( β ) φ ( α ) f ( x ) dx. But by noting H ( α ) = 0, the Fundamental Theorem of Calculus again gives H ( β ) = H ( β ) - H ( α ) = Z β α H 0 ( t ) dt = Z β α f ( φ ( t )) φ 0 ( t ) dt as desired. 2. Let f,g ∈ R [ a,b ]. (a) If t R , show that R b a ( tf ± g ) 2 0. (b) Use (a) to show that 2 | R b a fg | ≤ t R b a f 2 + (1 /t ) R b a g 2 for t > 0. (c) If R b a f 2 = 0, show that R b a fg = 0. (d) Now prove that ± ± ± R b a fg ± ± ± 2 ( R b a | fg | ) 2 ( R b a f 2 )( R b a g 2 ). This inequality is called the Schwarz inequality. Since t R , it follows that ( tf ± g ) is integrable for each t . By 7.3.16, (

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hw28solutions - Math 521 Advanced Calculus I Instructor J...

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